Method, apparatus, and computer program product for solving single and multiple integral equations using spreadsheet software

ABSTRACT

The present disclosure provides a method, an apparatus, and a computer program product which communicate with a programming interface of a spreadsheet software application for computing a value of a spreadsheet formula depending on at least one variable cell explicitly or implicitly through interdependence on additional nested formulas, for a supplied value of the variable cell, and without storing or modifying any stored value in the spreadsheet application. In one aspect, the present disclosure teaches a method for computing a numerical value of a single or multiple integral equation using a pure integration function of the spreadsheet application.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of application Ser. No. 15/861,897filed Jan. 4, 2018, which is a continuation of application Ser. No.15/003,848 filed Jan. 22, 2016, which is a continuation of applicationSer. No. 14/588,909, filed Jan. 3, 2015, the context of which areexpressly incorporated.

BACKGROUND Technical Field

This present disclosure relates to a method, an apparatus, and acomputer program product for solving equation system models usingspreadsheet software. More particularly, the present disclosure relatesto computer methods and apparatuses for computing numerical solutionsfor calculus problems in a spreadsheet application including problems ofintegration, differentiation, nonlinear systems, differential equationsand optimization.

Description of Related Art

Spreadsheet applications (such as Excel® of Microsoft Corp., hereafter“Excel”) are used universally for business as well as engineering andscientific data modeling and analysis. The widespread adoption of thespreadsheet application is supported in part by its inherent simplicityof defining formulas and manipulating data, in addition to a practicalset of built-in functions, charting tools, and extensibility. Numerousextensions (aka Add-Ins) have been developed to extend Excel utilityinto various applications, including solutions to differential equationsand optimization of algebraic and stochastic models. However,computational problems in constrained optimization involving systems ofdifferential and differential-algebraic equations are beyond thelimitations of existing Add-Ins. Such problems arise frequently inengineering and scientific applications when it is required to computeoptimal values for a parameterized system of differential equations tosatisfy objective constraints on the system response. Prior art Add-Insdo not offer an integrated solution to this class of problems in Excelspreadsheet application.

On the other hand, constrained optimization involving algebraicobjective function and constraints has long been supported in Excel. Infact, Excel ships with a standard built in solver that is inherentlylimited to algebraic formulas that can be defined in Excel worksheets.Excel solver is an instance of an Excel command that is executed via aninteractive dialogue, and performs operations in response to useractions as follows:

-   -   1. The user selects worksheet cells to hold initial values for        each decision variable in a model.    -   2. The user creates a formula in another worksheet cell that        calculates the objective function for the model.    -   3. The user creates additional formulas in worksheet cells to        calculate the left hand sides of each constraint.    -   4. Via the command interface dialogue, the user selects the        decision variables cells, the objective cell, the constraints        cells, and enters desired bounds for each constraint and        variable.    -   5. The solver iterates, altering the decision variables cells'        values and recalculating the dependent objective and constraints        cells values until such values are found which minimize the        objective function value and satisfy the constraints bounds.

When the solver completes, it will have overwritten the values of theinput cells by the final results values.

A command is the standard mechanism utilized by an Add-In for evaluatingformulas in Excel. Values for the independent cells are changed and thedependent formulas cells are recalculated by the spreadsheet applicationengine. Several available Add-Ins related to optimization anddifferential equations problems utilize the command approach to extendExcel utility. Frontline Systems Inc. offers through its website,Solver.com, a suite of solvers for optimization problems that arelimited to algebraic and stochastic models. In contrast, solvers fordifferential equations in Excel have been rather limited. PolymathSoftware claims on their website, Polymath-Software.com, that theirODE_Solver Add-In to Excel is the only available solver for ordinarydifferential equations in Excel. Polymath ODE_Solver is described byShacham et al in the paper: “Closing the Gap between Numerical SoftwarePackage and Spreadsheet Users in Process Computations” published in theproceedings ASEE Annual Conference, 2005 and is available online atPolymath-Software.com. ODE_Solver is based on the command approach, andis incapable of supporting constrained optimization on the differentialequations. Joseph Billo, in his book titled: “Excel for Scientists andEngineers”, WILEY-INTERSCIENCE 2007, describes several command-basedVisual Basic for Application (VBA) recipes to solve differentialequations models in Excel. Similar to Polymath solver, these methodsrely on the command approach for evaluating system formulas and are onlysuitable for simulating standalone, pre-configured differentialequations, but are incapable of supporting constrained optimization ofsuch systems.

An Excel command behavior differs fundamentally from a spreadsheetfunction because it mutates its inputs, and can trigger additionalactions. Furthermore, as described in the article “Excel Commands,Functions, and States” which is available online from Microsoft MSDNlibrary, commands cannot be invoked programmatically from spreadsheetfunctions, and they are never called by Excel during a recalculation.Commands are designed as interactive tools, and are not intended to beemployed programmatically as re-usable functions in a tiered functionalprogram. These restrictions limit the scope of problems that can besolved using the command mechanism.

On the other hand, the alternative mechanism for extending Excel'sutility is through the addition of new spreadsheet functions. Excelpermits only pure function behavior, and unlike commands, grantsfunctions restricted access to its features through its AdvancedPrograming Interface (API). A pure function cannot mutate its input orany other value in Excel. It cannot evaluate formulas, and it isrestricted to operating on constant inputs. Some external programs, suchas MATLAB (distributed by MathWorks Inc.), offer interfaces to Excel toexpose a portion of their functionality as extended Excel functions.Here, Excel primarily serves as a platform for inputting data through aproxy function while computations are performed utilizing the externalprogram. This model permits exchange of basic data types such asnumbers, but cannot be used to expose differential and optimizationsolvers, which require the ability to exchange and evaluate functions.

Whether utilizing the command or the function mechanisms for extendingExcel spreadsheet utility, prior art has been confined to the inherentdesign limitations of either approach, and has been unable to extend thespreadsheet utility to compute optimal results for problems involvingconstrained optimization of parameterized differential equation systems.

SUMMARY

A computer method is developed for computing the value of an arbitraryformula in a spreadsheet application for provided values of one or moredesignated variable cells on which the formula depends explicitly orimplicitly, such that no data is modified in the spreadsheetapplication. This method is utilized to overcome the inherent designlimitations of the spreadsheet application, which restricts spreadsheetfunctions to operating on constant inputs.

Accordingly, several spreadsheet functions utilizing said method aredeveloped for computing properties and responses of systems modelled byone or more variables, dependent formulas, and parameters. Furthermore,an optimization process is developed which is carried out by spreadsheetfunctions utilizing said method, and arranged in a functional paradigmin a spreadsheet application. The process computes optimal parametersthat satisfy objective constraints on the response of a parameterizedsystem modelled by formulas, variables, and parameters in a spreadsheetapplication.

A plurality of novel spreadsheet functions and the optimization processutilizing said method are incorporated in a computer program whichextends Microsoft Excel spreadsheet application as an Add-In. Theprogram can be executed on any computer hardware configuration thatsupports running Excel spreadsheet application. Several illustrativeproblems are solved using said program and presented as examples.

BRIEF DESCRIPTION OF THE DRAWINGS

The figures include several flowcharts detailing the algorithmsdeveloped to support the disclosure, as well as additional illustrativeresults for the embodiments. Related flowcharts have been assigned thesame figure number but with different alphabetic suffixes.

FIG. 1 shows two flowcharts providing high-level descriptions forModules-D & E, which support the method for circumventing spreadsheetfunction limitations. Module-D has two processes 101 and 201 shown inthe figure by numerals.

FIGS. 2A and 2B show flowcharts for algorithms 101A and 101B, which areutilized by process 101 of Module-D in the construction of a graphrepresentation of a formula.

FIG. 2T shows flowchart for algorithm 101T which is utilized inalgorithms 101A of Module-D for tokenizing a formula expression.

FIGS. 3A and 3B show sample output graphs produced by algorithms 101Aand 101B of Module-D.

FIGS. 4A to 4C show flowcharts for algorithm 201A, 201B and its helperprocedure 201B-H, and 201C and its helper procedure 201C-H, which areutilized by process 201 of Module-D for further processing of the graphrepresentation.

FIGS. 5A to 5C show sample output graphs produced by algorithms 201A,201B, and 201C of Module-D.

FIG. 6 shows flowchart for algorithm 301, which is utilized by Module-Efor evaluating the graph produced by Module-D.

FIG. 7 shows flowchart for the main procedure used to integrateModules-D & E and a computer algorithm to develop an unconventionalspreadsheet function, which can receive and evaluate formulas to solve aproblem represented by formulas and variables.

FIG. 8A shows the results layout format produced by a spreadsheetfunction utilizing Modules-D & E to compute the response of an ordinarydifferential system of equations.

FIG. 8B shows the results layout format produced by a spreadsheetfunction utilizing Modules-D & E to compute the response of a partialdifferential system of equations.

FIGS. 9A to 9C show plots of results obtained by spreadsheet functionsutilizing Modules-D & E to solve examples of ordinary and partialdifferential systems of equations.

FIG. 10 shows flowchart for the algorithm utilized by criterionspreadsheet functions ODEVAL( ) and PDEVAL( ) described below, whichenable definition of constraints on the response of a differentialsystem for a three-step optimization process.

FIGS. 11A and 11B show plots for the initial and optimized responsesrespectively, of a 2^(nd) order dynamical system used in the firstexample below for illustrating the three-step optimization process.

FIG. 12 shows a schematic for the forces acting on a train which is usedin the second example below for illustrating the three-step optimizationprocess.

DETAILED DESCRIPTION

In order to develop a process for carrying out the constrainedoptimization of a parameterized system of differential equations in aspreadsheet application, it is imperative to adopt a functionalformulation. In a functional formulation, the response of thedifferential system is obtained by means of a spreadsheet function thatcan be evaluated repeatedly for arbitrary values of its inputsparameters as required by the optimization process. As pointed outearlier, a command is not employable as a re-usable function in a tieredfunctional formulation, because it cannot be invoked programmaticallyfrom spreadsheet functions. Furthermore, it lacks the requiredproperties of purity and recursion, which are attributes of functionsonly. A pure function does not mutate its inputs or produce any sideeffects, and evaluates to a deterministic value for a given input, whilea recursion property allows the function to invoke itself with differentinputs. Therefore, the inherent design limitations of the spreadsheetapplication, which restrict spreadsheet functions to operating onconstant inputs only, must be overcome to make a functional formulationfeasible.

Accordingly, a method is developed which enables a spreadsheet functionto receive and evaluate formulas, and not just constant inputs, whilepreserving its purity and recursive properties. The method represents abreakthrough in the utilization of the spreadsheet application, andforms the basis for the creation of new class of unconventionalspreadsheet functions that have never existed in a spreadsheetapplication before. The method is described next following a briefdescription of the spreadsheet application's basic concepts. Theprovided description is based on the Excel spreadsheet application (thatis, a native spreadsheet software program executable over an operatingsystem), although the same concepts apply to other spreadsheet programsincluding Google's spreadsheet (that is, a web-based spreadsheetsoftware program executable over a web browser software program).

Basic Spreadsheet Concepts

A worksheet in Excel is composed of a large structured grid. Each cellin the grid is referenced by its column label and row number (e.g., A1),and represents a global memory placeholder. A range of cells can bereferenced as a rectangular array (e.g., A1:B3) or a union of disjointarrays and cells (e.g., (X1, A1:A3)). A cell may store a constant valueor a formula defined using basic spreadsheet syntax (e.g.,‘=SQRT(X1{circumflex over ( )}2+Y1*Y1)’). The spreadsheet engine insuresorderly evaluation of all dependent formulas upon a change in the valueof any cell. A general function can thus be identified by a root formulaand a list of variable cells. Nested dependency allows arbitrarilycomplex functions to be constructed. To motivate the possibilities,consider the formula ‘=SUM(X1:Z1)’ assigned to A1, the pair (A1, Y1)identifies the function ƒ(y)=X1+y+Z, where X1 and Z1 are treated asconstant values. In another example, consider the formula ‘=1+COS(B1)’assigned to A1, and the formula ‘=SQRT(ABS(X1))’ assigned to B1, thepair (A1,X1) identifies the function ƒ(x)=1+cos(√{square root over(|x|)}).

Excel provides a standard mechanism for integrating a new spreadsheetfunction, and an Advanced Programming Interface (API) to enable theadded function to interact programmatically with its core operations andengine. Documentation of the API as well as associated softwaredevelopment kit are available online from Microsoft Office DeveloperCenter. Through the API, a spreadsheet function is permitted to do thefollowing:

-   -   Obtain the value of any cell.    -   Obtain the formula expression stored in any cell.    -   Evaluate a built in worksheet function by supplying its name and        values for its parameters.    -   Evaluate the value of a short formula expression provided the        expression length is within a specified limit. (Currently set at        255 characters.)

On the other hand, a spreadsheet function is not permitted by the APIto:

-   -   Change the value of any cell except for producing a result in        its own assigned memory.    -   Recalculate or evaluate any dependent formulas cells.

These two operations are permitted for commands only.

Method for Circumventing Spreadsheet Function Limitations

FIG. 1 shows a high level flowchart of a method designed to circumventspreadsheet restrictions, and enable a spreadsheet function to passivelyevaluate an arbitrary formula for any values of its list of variablecells, without modifying any data in the spreadsheet application. Themethod is comprised of two modules:

-   -   I. Module-D, which represents the formula as an expanded graph        of relational nodes containing evaluable expressions via the        Excel API, and    -   II. Module-E, which, given values for the variable cells,        evaluates the relational graph to compute the formula value.

In the following, a detailed description is provided for Modules-D & Eof the enabling method.

I. Description of Module-D: FIGS. 1 to 5C

Module-D, shown in FIG. 1, comprises processes 101 and 201, which areexecuted on a computer having a processor and memory, and configured torun Excel spreadsheet application:

-   1. Process 101: captures the definition of the formula from Excel    and constructs a relational graph of nodes representing its    inter-dependence on nested formulas.-   2. Process 201: expands the relational graph produced by process 101    through a sequence of decomposition steps to produce an evaluation    graph whose nodes contain evaluable expressions via the Excel API.    Process 201 also identifies special function calls in the graph to    support recursion.

Processes 101 and 201 employ a number of algorithms to accomplish theirtasks. These algorithms are described in detail next, starting withalgorithm 101T, which is utilized in process 101 to tokenize a formulainto a list of substring tokens.

Description of Algorithm 101T: FIG. 2T

An Excel formula can contain any number of the following basic types:Functions, References, Constants and Operators. An Excel formulaexpression can be decomposed into a sequence of substrings, (akatokens), such that each substring represents a logical building entityof the entire expression. Algorithm 101T uses consecutive ordered searchsteps to identify and collect the tokens shown in Table 1 in any givenformula expression.

TABLE 1 Token Type Significance FunctionOpen start of a function callFunctionClose closing parenthesis of a function call ParameterSeparatora comma separating function parameters SubExpressionOpen start of agrouped algebraic subexpression SubExpressionClose closing parenthesisof a grouped algebraic subexpression Reference Excel cell or rangereference Constant literals, numbers, and Excel Booleans and errorvalues ArrayConstant a list of numbers enclosed by curly brackets { }PreFixOperator unary −, + sign operator InFixOperator +, −, */, ^(∧), >,<, =, >=, <=, <>, & operators PostFixOperator % operator

FIG. 2T shows a flowchart for algorithm 101T ordered search steps. Aftereach search step, the collected tokens are inserted in a list sorted bythe token starting index in the formula expression 181, and thecorresponding substrings in the formula are masked so they are ignoredin the next search step of the formula 182. Staring from an inputformula expression 180:

-   -   1. Collect FunctionOpen tokens: use a regular expression        representing a legal name identifier followed immediately by “(”        183.    -   2. Collect FunctionClose tokens: search for a matching “)” for        each collected functionOpen token 184.    -   3. Collect ArrayConstant tokens: search for any text enclosed by        a pair of curly brackets{ } 185.    -   4. Collect Constant tokens: use a regular expression for numeric        constants; search for a pair of double quotes for literals;        search for Excel constants for Booleans and errors 186.    -   5. Collect operators (without classification yet): search for +,        −, * /, {circumflex over ( )}, >, <, =, >=, <=, < >, &, % 187.    -   6. Collect union references: search for pairs of matching        parenthesis that contain commas but no other parenthesis.        Collect each pair as a Reference token 188.    -   7. Collect all intersection references: search for pairs of        matching parenthesis that contain white spaces separating other        text but no commas or other parenthesis. Collect each pair as a        Reference token 189.    -   8. Collect SubExcpressionOpen tokens: search for a beginning “(”        190.    -   9. Collect SubExpressionClose tokens: search for a matching “)”        191.    -   10. Collect ParameterSeparator tokens: search for commas 192.    -   11. Collect each remaining substring among collected tokens as a        Reference token 193.    -   12. Classify collected operators tokens as pre-, in-, or        post-fix operators 194.

Algorithm 101T returns an ordered list of tokens for the input formulaexpression upon completion 195. For example, applying the aboveprocedure to the formula expression‘=−1+EXP((X1−1){circumflex over( )}2*SIN(Y1))+SUM(A1:C4,D1)+IF(P5>=0,P6,P7%)’produces the list of tokens shown in Table 2.

TABLE 2 Token No. Text Type 0 “=” Formula Start 1 “−” PreFix Operator 2“1” Constant 3 “+” InFix Operator 4 “EXP(” FunctionOpen 5 “(”SubExpressionOpen 6 “X1” Reference 7 “−” Infix Operator 8 “1” Constant 9“)” SubExpressionClose 10 “^(∧)” Infix Operator 11 “2” Constant 12 “*”Infix Operator 13 “SIN(” FunctionOpen 14 “Y1” Reference 15 “)”FunctionClose 16 “)” FunctionClose 17 “+” Infix Operator 18 “SUM(”FunctionOpen 19 “A1:C4” Reference 20 “,” ParameterSeparator 21 “D1”Reference 22 “)” FunctionClose 23 “+” Infix Operator 24 “IF(”FunctionOpen 25 “P5” Reference 26 “>=” Infix Operator 27 “0” Constant 28“,” ParameterSeparator 29 “P6” Reference 30 “,” ParameterSeparator 31“P7” Reference 32 “%” PostFix Operator 33 “)” FunctionCloseDescription of Process 101 of Module-D: FIGS. 2A to 3B

Process 101 employs algorithms 101T, 101A and 1018 to capture thedefinition of a formula from Excel, and construct a relational graph ofnodes representing its dependence on nested formulas. FIG. 2A shows theflowchart of algorithm 101A which takes as inputs a root formula celland a list of variable cells 125. It obtains the cell formula expressionusing Excel API 126, and creates a root node 127 representing the rootformula, and then executes a recursive procedure that performs thefollowing steps on the current node 128: it gets the recorded formulaexpression in the node 129, and applies algorithm 101T to tokenize theformula, then records the tokens in the node 130. Next, it collects thelist of references in the tokens 131. If the list is empty 132, it endsprocessing for the current node and returns 133. Otherwise, it iteratesover the references list 134 and performs either a) or b) belowdepending whether the reference is a multi-cell range or a single cell135:

-   a) If the reference represents a multi-cell range, algorithms 101A    inserts a child node of type R which records the multi-cell range ID    136. Using Excel API, it obtains the list of cells in the range 137    and iterates through each cell to perform the following steps 138:    If the cell ID matches any of the variable cells 139, it inserts a    child node of type V which records the variable cell ID 140, then    continues to process the next cell 138. Otherwise, if the cell    contains a formula 141, it obtains the formula expression using    Excel API 142, and inserts a child node of type C which records the    cell ID and its formula expression 143. The Procedure then calls    itself recursively to process the new child C node it just created    144 or returns when it completes iterating through the cells in the    range 145.-   b) If the reference represents a single cell 135 and it matches any    of the variable cells 146, algorithm 101A inserts a child node of    type V which records the variable cell ID 147, then continues to    process the next cell in the list 134. Otherwise, if the cell    contains a formula 148, it obtains its expression from Excel API    149, and inserts a child node of type C, which records the cell ID    and the formula 150, then calls itself recursively to process the    new child node it just created 151, or returns when it completes    iterating through the references list 152.

FIG. 3A shows a sample graph outcome at the completion of algorithm101A. The graph contains any number of three types of nodes: C, R and V.Nodes of type C contain a cell ID, a tokenized formula expression, andany number of child nodes of types V, C and R. Nodes of type R contain amulti-cell range ID, and any number of child nodes of types C and V.Nodes of type V contain a variable cell ID and can only exist as leafnodes.

The second step of Process 101 applies algorithm 101B to the graphproduced by algorithm 101A. FIG. 2B shows the flowchart for algorithm101B, which employs a recursive procedure that traverses the graph anddeletes nodes of types C or R with no immediate or remote children nodesof type V, while setting up interdependence information for retainednodes. Starting at the root node of a graph with unvisited nodes 160,recursive algorithm 101B first checks if the current node is the rootnode and has all its children visited 161. If true, algorithm 101Bterminates processing and returns 162. Otherwise, depending on the nodetype, algorithm 101B performs any of the following procedures a), b) orc):

-   -   a) If the current node is a V node 163, it creates a list of        dynamic links to instances of the V node's cell ID in the parent        node's tokenized formula expression, and marks the V node as        visited 164. It then ends processing for the current V node and        returns to the next sibling or parent 165.    -   b) If the current node is an R node 166 with no children 167, it        deletes it 168, ends processing for the current R node and        returns to the next sibling or parent 169. Otherwise, if the R        node has children 167, it performs the following steps on the R        node:        -   1. It defines an equivalent union reference representation            for the multi-cell range node 170. The union reference is            defined as the union of the immediate child node cells and            the remaining non-child cells in the range.        -   2. It tokenizes the union reference expression and records            it as the R node's formula 170.        -   If the R node has unvisited children 171, algorithm 101B            obtains the list of immediate child nodes for the R node            172, iterates through the child node list 173, and calls            itself recursively for each child node 174. When algorithm            101B completes iterating through the child node list, it            ends processing for the current R node and returns to the            next sibling or parent 175.    -   c) If the current node is a C node and has unvisited children        171, algorithm 101B obtains the list of immediate child nodes        for the C node 172, iterates through the child node list 173,        and calls itself recursively for each child node 174. When        algorithm 101B completes iterating through the child node list,        it ends processing for the current C node and returns to the        next sibling or parent 175.

Algorithm 101B repeats until it reaches a leaf node without unvisitedchild nodes 171. If the leaf node does not have child nodes of type V176, it deletes the leaf node 177 and returns to the next sibling orparent 178. Otherwise, it performs the following steps on the leaf node:

-   1. It creates a list of dynamic links in the node to instances of    the node's cell ID in the parent node's tokenized formula expression    179.-   2. It marks the node as visited, reducing the count of unvisited    nodes for the parent node 180.

Algorithm 101B then ends processing for the current leaf node andreturns to the next sibling or parent 178.

Algorithm 101B terminates when it returns to the root node aftervisiting all nodes in the graph. FIG. 3B shows a possible outcome ofalgorithm 101B when applied to the graph shown in FIG. 3A. FIG. 3B alsoshows the recorded information in the different types of nodes in thegraph at the completion of process 101.

Description of Process 201 of Module-D: FIGS. 4A to 5C

Process 201 employs algorithms 201A, 201B and 201C to expand therelational graph produced by process 101 into an evaluation graph whosenodes contain expressions evaluable via the Excel API. FIG. 4A shows theflowchart for algorithm 201A, which visits every node of type C in thegraph and attaches a sub graph of type F nodes representing thehierarchy of function calls present in the formula expression of thenode. From the tokenized formula of the current C node 225, algorithm201A collects and iterates over the first-level function callexpressions present in the formula 226. It inserts a node of type F foreach first-level function call expression 227, and defines a formula forthe F node as a dynamic link to the instance of this function expressionin the tokenized formula of the parent node 228. Algorithm 201A thencalls itself recursively to process the formula of the newly insertedchild F node 229. It returns when there are no more first-level functionexpressions in the current node's tokenized formula 230. FIG. 5A shows apossible expanded output graph of algorithm 201A when applied to thegraph shown in FIG. 3B. Algorithm 201A serves two purposes:

-   -   1. It helps reduce the overall formula expression length in a        parent node to meet the length limit requirement of the Excel        API. This is because each extracted nested function call        expression is replaced in the parent node formula by its shorter        equivalent value during evaluation. However, the graph may still        contain nodes of type C or F that are longer than the admissible        length for the Excel API. These nodes are further processed by        algorithms 201B and 201C described next.    -   2. It isolates all nested functions calls present in the        formulas of the graph. This is required to identify and handle        special recursive function calls present in the graph during the        evaluation, which is described afterwards.

Algorithms 201B and 201C are utilized next for further processing of thenodes' expressions to meet Excel API expression evaluation length limit,LENMAX. Algorithm 201B is shown in the flowchart of FIG. 4B. It regroupsevery infix operator and its two operands into a subexpression, whilerespecting order precedency of the operators. Starting with a tokenizedinput expression 240, algorithms 201B uses recursion to locate anypre-existing subexpression in the input whose length exceeds LENMAX. Asubexpression is a sequence of tokens starting with a token of typeSubExpressionOpen and ending with a token of type SubExpressionClose.Algorithm 201B collects and iterates over the first-level subexpressionspresent in the tokenized input expression 241, and calls itselfrecursively to process any first-level subexpression, sexpr, whoselength exceeds the Excel API length limit, LENMAX 243. When no moresubexpressions are found in the current input expression, algorithm 201Bapplies procedure 201B-H to break up the current input expression 244,then terminates processing of the current input expression and returnsfrom the current recursive call 245.

Helper procedure 201B-H, which is shown in FIG. 4B, searches its inputfor the next infix operator in order of precedence, and regroups it withits left and right operands into a new subexpression. An operand may bea reference, a subexpression, or function call. Procedure 201B-H ignoresa regrouped subexpression in its searches and treats it as an operand.Starting with an input list of expression tokens 246, procedure 201B-Hsearches for any exponentiation infix operator 247, and groups it withits left and right operands into a new subexpression 248. Procedure201B-H repeats the same process for the remaining infix operators in thefollowing order: multiplication, division, addition, subtraction, andlogical as shown in the flowchart of FIG. 4B. When all infix operatorspresent in the input tokens have been grouped with their operands,procedure 201B-H terminates and returns the updated list of regroupedtokens 249.

As an example, algorithm 201B transforms the following input expressionA+B*C*D−(E−F*G){circumflex over ( )}(H−1)+J*K/L−Minto(((A+((B*C)*D))−((E−(F*G)){circumflex over ( )}(H−I)))+(((J*K)/L)−M)).

Algorithm 201B is utilized by algorithm 201C for representing longexpressions in the graph nodes by an equivalent sequence of evaluablesubexpressions. FIG. 4C shows the flowchart for algorithm 201C which isapplied to every node of type C or F in the graph. Algorithm 201Cattaches an evaluation sub graphs of type E nodes to any C or F nodewhose formula expression length exceeds the Excel API evaluation limit,LENMAX. Starting at a node with tokenized formula expression 270, if theexpression length is within the Excel API evaluation limit 271,algorithm 201C terminates processing of the node and returns 272.Otherwise, if the node type is C 273, algorithm 201C applies algorithm201B to regroup the node's formula into a sequence of binary arithmeticsubexpressions 274, followed by applying procedure 201C-H to the node'sregrouped formula to attach the sub graph of type E nodes 275. Algorithm201C then ends processing of the current node and returns 276. On theother hand, if the node type is F 273, algorithm 201C iterates over thefunction parameters 277, and inserts a new node of type E 279 for anyfunction parameter whose expression length is greater than itsequivalent value length, RESELN 278. Algorithm 201C then defines the Enode's formula as a dynamic link to the instance of this parameter inthe tokenized formula of the parent node 280. It then applies algorithm201B on the newly created E node to regroup its expression into asequence of binary arithmetic subexpressions 281, followed by applyingprocedure 201C-H to attach a sub graph 282. Algorithm 201C then eitherprocesses the next parameter of the function 277, or returns if thereare no more parameters 283.

Helper procedure 201C-H is shown FIG. 4C. Starting with an inputtokenized formula expression 285, procedure 201C-H collects and iteratesover the first-level subexpressions present in the input expression 286.It inserts child node of type E 288 for each first-level subexpression,sexpr, longer than its equivalent value length, RESLEN 287, and definesits formula as a dynamic link to the instance of sexpr in the tokenizedformula of the parent node 289. If the length of sexpr exceeds the ExcelAPI limit, LENMAX 290, procedure 201C-H calls itself recursively on theE node to process any nested subexpressions within sexpr 291. Procedure201C-H returns from the current recursive call 292 when there are nomore subexpressions in the input expression, or when all subexpressionsare within admissible length.

FIG. 5B shows a sub graph of type E nodes for the regrouped example(((A+((B*C)*D))−((E−(F*G)){circumflex over ( )}(H−I)))+(((J*K)/L)−M)).FIG. 5C shows a possible outcome of algorithm 201C when applied to thegraph shown in FIG. 5A, along with the recorded data in the F and Enodes at the completion of process 201.

The actual structure of an evaluation graph produced by Module-D dependson the complexity of the formula it represents. It is still possiblethat output of Module-D may contain nodes of type F with functionsexpressions that are irreducible below the Excel API evaluation limit,due to too many parameters or an expanded long union referenceparameter. Since a function expression must be evaluated in whole toform a valid evaluable expression, Excel API expression evaluationmethod cannot be used to evaluate such long function expressions. Thisspecial case is handled by a sub process part of Module-E, using analternative Excel API method that computes the function directly bysupplying its label identifier and list of its parameter values.

II. Description of Module-E: FIG. 6

Module-E is executed on a computer having a processor and memory andconfigured to run Excel spreadsheet application. It computes the valueof the formula represented by the evaluation graph produced by Module-D,for the given values of the variable cells. Module-E employs algorithm301, which traverses the tree structure, visiting child nodes in aprescribed order until a leaf node is reached. It computes its value,updates its parent formula expression, and drops the leaf node (i.e.,mark it as visited). When a parent node loses all its child nodes, itbecomes a leaf itself. The process is repeated until all leafs have beendropped and the root parent node is reached. Computing the value of theroot node formula expression produces the final value of the formularepresented by the graph.

FIG. 6 shows the flowchart for algorithm 301, which starts at the rootnode of an evaluation graph with unvisited nodes 350. It iteratesthrough the unvisited child nodes lists of the current input node in thefollowing order: V 351, C 352, R 353, F 354, and E 355 and calls itselfrecursively for each child node in a given list 356 until it reaches aleaf node without any unvisited child nodes 357. When algorithm 301completes iterating through all child nodes of the current input node,it calls itself recursively on the current input node 358 which becomesa leaf itself after all its children have been marked visited.

If the leaf node is a V node 359, algorithm 301 records the providednumerical value of the variable cell ID in the node's list of dynamiclinks 360. It marks the V node as visited 361 and returns from therecursive call to the next sibling or parent node 362. Otherwise, if theleaf node is an R node 363, algorithm 301 records the expanded unionreference expression of the R node in its list of dynamic links 364. Itmarks the R node as visited 361, and returns from the recursive call tothe next sibling or parent node 362.

On the other hand, if the current node is a C or F node, algorithm 301forms the expression of the node's formula from its list of dynamic linktokens while replacing any reference, function call, or subexpressioncontaining recorded result values by its value expression 365. Thedesign of the evaluation graph and the order of child nodes evaluationsensure that the formula expression formed for a type C or E leaf node isevaluable by the Excel API evaluation method. However for a type F leafnode, the formula which contains a function call may potentially exceedthe Excel API length limit for expression evaluation, or it may containa recursive special function call 366. Such an F node is evaluated usingprocedure 301F which handles both long and special function calls 367.Special functions are not part of the standard built-in library ofworksheet functions in Excel. They are part of the Add-In to Excel, andinclude a set of computational functions described in the followingsection. These Add-In functions have available procedures that can beexecuted directly by Module-E by supplying their current parameters inthe graph. This achieves recursion naturally in the event the rootformula contains an Add-In function call that is referenced again in anested graph node. On the other hand, a long function call that is partof the standard Excel library of worksheet functions is evaluated bymeans of the Excel API by providing the function name identifier and thelist of values for its parameters. On the other hand, a regular C, E orF node is evaluated directly by means of the Excel API expressionevaluation method 368.

If the current leaf node is not the root node 369, algorithm 301converts the evaluated node result to textual expression 370, andperforms one of three steps:

-   -   If the leaf node type is C 371, it records the result into its        list of dynamic links to instances of the node's cell ID in the        parent's formula tokens 372. During the formation of the        parent's formula expression, instances of this cell ID are        replaced by their stored value.    -   If the leaf node type is F 373, it records the result in the        FunctionOpen token of the node's formula 374. During the        formation of the parent's formula expression, the entire        function expression will be replaced by the recorded result.    -   If the leaf node type is E 375, it records the result in the        SubExpressionOpen token in the node's formula tokens 376. During        the formation of the parent's formula expression, the entire        subexpression will be replaced by the recorded result.

Since each child node's formula expression is defined by dynamic linksto its parent's formula tokens, the result of a leaf node evaluation isautomatically seen by the parent and siblings. Algorithm 301 then marksthe leaf node as visited 377 and returns from the recursive call to thenext sibling or parent node 378.

Algorithm 301 terminates when the current leaf node is the root node ofthe graph 369. It returns the evaluated node result, which representsthe value of the formula represented by the graph 379.

The merits of the evaluation graph design are evident in the efficientand systematic incremental evaluation of the graph nodes leading to thefinal value of the root formula it represents. In this design, I haverelied often on recursive procedures to perform the logic of theprocess, although alternative non-recursive procedures are alsopossible. Furthermore, alternative graph representations are alsopossible using other techniques such as S-Expressions binary trees.Other techniques are also possible for tokenizing a formula expressionsuch as relying on grammar rules using BNF (Backus-Naur Form) orextended BNF in conjunction with a Lexer. I have also utilized the ExcelAPI as the primary engine to evaluate the sequence of evaluableexpressions in the graph nodes since it provides a natural, efficientand a readily available means for evaluating the expressions. Othermeans for evaluating the expressions are also possible and may benecessary in a spreadsheet application whose API does not provide anexpression evaluation method. Such means may include, among otherpossible techniques, converting an expression to a lambda expression, oranonymous function, and utilizing a programming language environmentsupporting evaluations of lambda expressions. However, alternativerepresentations, algorithms, or procedure implementations may differ incomplexity but serve the same objective: the passive evaluation of anarbitrary formula in a spreadsheet for any provided values of itsvariable cells by means of a method executed in a separate memorylocation of the computer, with no side effects in the spreadsheetapplication.

Novel Spreadsheet Computational Functions

Modules-D & E enable the creation of a new class of unconventionalcomputational spreadsheet functions that have never existed before in aspreadsheet application for solving a variety of complex models. UsingModules-D & E, a spreadsheet function can be developed to compute theresponse or property of any system which can be modelled by formulas,and whose response or property can be computed by means of an algorithmrequiring a finite number of evaluations of the formulas. FIG. 7 shows aflowchart for the systematic procedure required to develop such aspreadsheet function. The spreadsheet function is passed parametersdescribing a problem model that includes formulas, designated variables,and parameters 700. To solve the problem, it executes the followingsteps:

-   -   1. Utilizes Module-D to construct evaluable graphs for the        formulas 701.    -   2. Passes the model description and parameters to the        appropriate integrated solver algorithm 702.    -   3. Starts solver 703, and utilizes Module-E to evaluate the        formulas' graphs 705, and provides values to the solver whenever        the solver provides values for the variables and requests values        for the formulas 704.    -   4. Repeats step 3 until the solver computes a result 706.    -   5. Returns the result to the spreadsheet application 707.

The above procedure forms the basis for several developed spreadsheetfunctions incorporated in a computer program which integrates with Excelspreadsheet application as an Add-In. The spreadsheet functions aredescribed below, and form the pillars of a functional formulation for athree-step constrained optimization process described afterwards. Thespreadsheet functions are organized into three groups:

-   -   I. Utility functions for evaluating integrals and derivatives of        general formulas.    -   II. Differential-algebraic systems solvers.    -   III. Algebraic systems of equations and inequalities solver.

Using the Add-In program, several examples are presented whichillustrate the utilization of the spreadsheet functions in Excel.

I. Utility Functions:

a) Integration

The provided Excel spreadsheet function

-   -   QUADF(f, x, a, b, [options])        computes a general proper or improper integral of the form:

I = ∫_(a)^(b)f(x)dx

The parameters for QUADF( ) are described below:

-   1. f: cell reference for the formula of the integrand.-   2. x: cell reference for the variable of integration.-   3. a: the lower limit of integration which may be specified as    infinite.-   4. b: the upper limit of integration which may be specified as    infinite.-   5. [options]: a list of optional parameters for integration method    selection and control.

Based on the flowchart of FIG. 7, QUADF( ) uses Modules-D & E inconjunction with the QUADPACK library of fixed-order and adaptiveintegration routines described in the book: “QUADPACK A subroutinepackage for automatic integration”, Springer Verlag, 1983. The use ofQUADF( ) is demonstrated in the computation of the following integral inExcel:

$I = {{\int_{0}^{1}{\frac{\ln\mspace{11mu} x}{\sqrt{x}}{dx}}} = {- 4}}$

The integrand formula is defined in cell A1 using cell X1 as variable.Evaluating QUADF( ) formula in cell A2 computes the integral result asshown in Table 3.

TABLE 3 A 1 =LN(X1)/SQRT(X1) 2 =QUADF(A1, X1, 0, 1) → 4

Similarly, Table 4 illustrates using QUADF( ) to compute the improperintegral with infinite limits:

$I = {{\int_{- \infty}^{\infty}{e^{- x^{2}}{dx}}} = \sqrt{\pi}}$

TABLE 4 A 1 =EXP(−X1 * X1) 2 =QUADF(A1, X1, “−INF”, “INF”) → 1.772453851b) Multiple Integrals

Taking advantage of the recursion support by Modules-D &E, QUADF( ) canbe applied recursively to compute multiple integrals of any order, suchas the triple integral:

$V = {{{\int_{0}^{2}{{dx}{\int_{0}^{3 - {\frac{3}{2}x}}{{dy}{\int_{0}^{6 - {3x} - {2y}}1}}}}} - {x\ {dz}}} = 3}$

The computation in Excel is illustrated in Table 5. Using X1, Y1 and Z1as variables, the integrand formula is defined in A1, and the formulasfor the variable upper integration limits are defined in B1 and B2. Theinner, middle, and outer integrals formulas are inserted in A2, A3 andA4 respectively, with each inner QUADF( ) formula serving as integrandfor the next outer QUADF( ) formula. Evaluating the outer integral in A4computes the triple integral value.

TABLE 5 A B 1 =1 − X1 =6 − 3 * X1 − 2 * Y1 2 =QUADF(A1, Z1, 0, B1) =3 −3 * X1/2 3 =QUADF(A2, Y1, 0, B2) 4 =QUADF(A3, X1, 0, 2) → 3c) Differentiation

The provided Excel spreadsheet function

-   -   DERIVF (f, x, p, n, [options])        computes the n^(th) derivative of function ƒ(x) at a point p:

${{d = {\frac{d^{n}}{{dx}^{n}}{f(x)}}}}_{p}$

The parameters for DERIVF( ) are described below:

-   1. f: cell reference for the formula to differentiate.-   2. x: cell reference for the variable of differentiation.-   3. p: value for the point of differentiation.-   4. n: the order of the derivative. Default value is one.-   5. [options]: a list of optional parameters for differentiation    method selection and control.

Based on the flowchart of FIG. 7, DERIVF( ) uses Modules-D & E inconjunction with Ridders' algorithm described in the book: “NumericalRecipes in C: The Art of Scientific Computing”, Cambridge UniversityPress, 1992. Ridders' algorithm produces superior accuracy compared to asimple finite difference scheme. DERIVF( ) is illustrated in Excel bycomputing the following example:

${{d = {\frac{d}{dx}\frac{\ln\mspace{11mu} x}{\sqrt{x}}}}}_{0.5} \cong 3.808685268$

The formula to differentiate is defined in cell A1 using X1 as variable.Evaluating DERIVF( ) formula in cell A2 computes the function derivativeat 0.5 as shown in Table 6.

TABLE 6 A 1 =LN(X1)/SQRT(X1) 2 =DERIVF(A1, X1, 0.5) → 3.808685268d) Partial Derivatives

Similar to QUADF( ), DERIVF( ) can be applied recursively in Excel tocompute higher order partial derivatives as illustrated by the followingexample:

${{d = {\frac{\partial}{\partial y}\frac{\partial}{\partial x}{\cos({xy})}}}}_{({\pi,\pi})} \cong 9.3394486379$

The formula to differentiate is defined in A1 using X1, Y1 as variables.The inner partial derivative is defined in A2, and serves as thedifferentiable formula for the outer partial derivative defined in A3.Evaluating A3 computes the result as shown in Table 7.

TABLE 7 A 1 =COS(X1 * Y1) 2 =DERIVF(A1, X1, PI( )) 3 =DERIVF(A2, Y1, PI()) → 9.3394486379II. Differential-Algebraic Equations Solversa) Initial Value Problems

The provided Excel spreadsheet function

-   -   IVSOLVE (sys, vars, interval, [options])        computes the response of an initial value ordinary        differential-algebraic system represented in the following        ordered form:

${\frac{{du}_{i}}{dt} = {f_{i}\left( {t,u,y} \right)}},{i = 1},n$0=g _(j)(t,u,y),j=1,m

with initial conditions u_(i)(0)=a_(i), y_(j)(0)=b_(j), and over thetime Interval t ∈ [0 T]. The parameters for IVSOLVE( ) are describedbelow:

-   1. sys: ordered list of (n+m) cells for the system formulas ƒ_(i)    and g_(j).-   2. vars: a list of (1+n+m) cells for the system variables {t, u, y}.    The cells for u=[u₁, u₂, . . . , u_(n)] and y=[y₁, y₂, . . . ,    y_(m)] must define the initial conditions for the variables.-   3. interval: the time interval limits, and control of output time    reporting as described in Table 8.-   4. m: the number of algebraic equations in the system. Default is    zero.-   5. [options]: a list of optional parameters for custom control. They    include supplying an analytic system Jacobian matrix, algorithm    selection, tolerances values, and custom algorithm settings.

TABLE 8 Format Output {t_initial, t_final} Output divided uniformly inallocated range {t_initial, t_final, ndiv} Output divided uniformly intondiv intervals {t0, t1, . . . , t_final} Output reported at custompoints

Based on the flowchart of FIG. 7, IVSOLVE( ) uses Modules-D & E forevaluation of the system formulas as requested by the underlining solveralgorithm. Its solver is based on RADAU5 described by Hairer and Wannerin the book: “Solving Ordinary Differential Equations II: Stiff andDifferential-Algebraic Problems”, Springer Series in ComputationalMathematics, 1996. IVSOLVE( ) is executed as an array formula in anallocated range in Excel. It evaluates to an ordered tabular array orresults with a layout described in FIG. 8A. In the results table, valuesof the system variables are reported at uniform or custom values of thetime variable and arranged in a column format. Control for the timevariable output is enabled in the third parameter as shown in Table 8.

The use of IVSOLVE( ) in Excel is illustrated by computing the responseof the following initial value differential-algebraic system:

$\frac{{du}_{1}}{dt} = {{{- 0.04}u_{1}} + {10^{4}u_{2}y_{1}}}$$\frac{{du}_{2}}{dt} = {{0.04u_{1}} - {10^{4}u_{2}y_{1}} - {3*10^{7}u_{2}^{2}}}$0=u ₁ +u ₂ +y ₁−1

with initial Conditions u₁(0)=1, u₂(0)=0, and y₁(0)=0 over the intervalt ∈ [0 1000].

Using cells T1, U1, U2, and U3 to represent system variables t, u₁, u₂,y₁ respectively, the system formulas are defined in cells A1:A3, andinitial conditions in cells U1:U3 as shown in Table 9.

TABLE 9 A U 1 =−0.04 * U1 + 10000 * U2 * U3 1 2 =0.04 * U1 − 10000 *U2 * U3 − 30000000 * U2{circumflex over ( )}2 0 3 =U1 + U2 + U3 − 1 0

Next, the array formula ‘=IVSOLVE (A1:A3, (T1,U1:U3), {0,1000}, 1)’ isevaluated in allocated Excel range B1:E22. It populates the range by theresult shown in Table 10. FIG. 9A shows a plot for the computed responseusing Excel charts.

TABLE 10 B C D E 1 T1 U1 U2 U3 2   0 1 0 0 3  50 0.69288 8.34415E−060.307111 4  100 0.617245 6.15388E−06 0.382748 5  150 0.5702295.12407E−06 0.429765 6  200 0.535843 4.48766E−06 0.464152 7  2500.508693 4.04187E−06 0.491303 8  300 0.48625 3.70564E−06 0.513746 9  3500.467146 3.44016E−06 0.532851 10  400 0.450524 3.22292E−06 0.549472 11 450 0.435837 3.04101E−06 0.56416 12  500 0.422673 2.88523E−06 0.57732413  550 0.41078 2.75007E−06 0.589217 14  600 0.399933 2.63133E−060.600064 15  650 0.38996 2.52558E−06 0.610037 16  700 0.3807532.43082E−06 0.619245 17  750 0.372209 2.34522E−06 0.627789 18  8000.364236 2.2674E−06 0.635761 19  850 0.356763 2.19606E−06 0.643235 20 900 0.349743 2.13047E−06 0.650255 22  950 0.343131 2.06992E−06 0.65686722 1000 0.336882 2.01377E-06 0.663116b) Boundary Value Problems

The provided Excel spreadsheet function

-   -   =BVSOLVE (sys, vars, pts, bcs, domain, m, [options])        computes the response of a multipoint boundary value ordinary        differential-algebraic system represented in the following        ordered form:

${\frac{{du}_{i}}{dx} = {f_{i}\left( {x,u,y} \right)}},{i = 1},n$0=g _(j)(x,u,y),j=1,m

with boundary conditions 0=bc_(i)(u, x) at x=p_(i) in the domain x ∈ [0L]. The parameters for BVSOLVE( ) are described below:

-   1. sys: ordered list of (n+m) cells for the system formulas ƒ_(i)    and g_(j).-   2. vars: a list of (1+n+m) cells for the system variables {x, u, y}.    The cells for u=[u₁, u₂, . . . , u_(n)] and y=[y₁, y₂, . . . ,    y_(m)] may define optional initial guess formulas for the variables.-   3. pts: locations of (n) boundary conditions points.-   4. bcs: a list of (n) cells for the corresponding boundary    conditions formulas.-   5. domain: the spatial interval limits, and control of output    spatial points similar to the format shown in Table 8.-   6. m: the number of algebraic equations in the system. Default is    zero.-   7. [options]: a list of optional parameters for custom control. They    include supplying an analytic system and boundary conditions    Jacobian matrices, algorithm selection, tolerances values, and    custom algorithm settings.

BVSOLVE( ) is similar to IVSOLVE( ) and evaluates to an identicalresults layout as shown in FIG. 8A, except that it solves a boundaryvalue problem based on the collocation algorithm described by Ascher andSpiteri in the paper: “Collocation software for boundary valuedifferential-algebraic equations, SIAM Journal on Scientific Computing”,1994 15, 938-952. The use of BVSOLVE( ) in Excel is illustrated bycomputing the response of the following boundary value problem:

$\frac{{du}_{1}}{dx} = u_{2}$$\frac{{du}_{2}}{dx} = {20\left( {{{- e^{u_{1}}}u_{2}} + {\frac{1}{2}{{\pi sin}\left( {\frac{1}{2}\pi\; x} \right)}e^{2u_{1}}}} \right)}$with boundary conditions: u₁(0)=0, u₁(1)=0 in the domain x ∈ [0 1]

Using cells X1, U1, U2 for system variables x, u₁, u₂ respectively, thesystem formulas are defined in cells A1:A3, the boundary conditionspoints in B1:B2, and the corresponding boundary conditions formulas incells C1:C2 as shown in Table 11.

TABLE 11 A B C 1 =U2 0 =U1 2 =20 * (−EXP(U1) * U2 + 0.5 * PI( ) *SIN(0.5 * 1 =U1 PI( ) * X1) * EXP(2 * U1))

Next, the array formula ‘=BVSOLVE (A1:A2, (X1,U1:U2), B1:62, C1:C2,{0,1})’ is evaluated in allocated Excel range G1:I23. It populates therange with the result shown in Table 12. FIG. 9B shows a plot of thedata using Excel charts.

TABLE 12 G H I 1 X1 U1 U2 2 0   0 −9.22395 3 0.05 −0.3043492 −3.93625 40.1 −0.44424441 −1.92551 5 0.15 −0.51353621 −0.95066 6 0.2 −0.54647914−0.41601 7 0.25 −0.55866051 −0.09636 8 0.3 −0.55796756   0.110589 9 0.35−0.5486075   0.256436 10 0.4 −0.53287428   0.368873 11 0.45 −0.51201677  0.463304 12 0.5 −0.48669581   0.548502 13 0.55 −0.45723449   0.62956114 0.6 −0.42375759   0.709506 15 0.65 −0.38627075   0.790212 16 0.7−0.34470297   0.872914 17 0.75 −0.29893093   0.958517 18 0.8 −0.24879048  1.047768 19 0.85 −0.19408162   1.14137 20 0.9 −0.13456902   1.24004221 0.95 −0.06997994   1.344568 22 1 −1.9429E−16   1.455833 23 Rel.Errors   1.87103E−06   8.35E−05c) Partial Differential Problems

The provided Excel spreadsheet function

-   -   PDSOLVE (sys, vars, lbcs, rbcs, domain, interval, [options])

Computes the response of an initial boundary-valuedifferential-algebraic system represented in the following ordered form:

${\frac{\partial u_{i}}{\partial t} = {f_{i}\left( {t,x,u,u_{x},u_{xx}} \right)}},{i = 1},n$with initial conditions u_(i)(t, x)=0 at t=0, and boundary conditionsbc_(i)(x, u, u_(x))=0 at x=0 and L, in the time interval: t ∈ [0 T] andspatial domain: x ∈ [0 L]. The parameters for PDSOLVE( ) are describedbelow:

-   1. sys: ordered list of (n) cells for the system formulas ƒ_(i).-   2. vars: a list of (2+3n) cells for the system variables {t, x, u,    u_(x), u_(xx)}. The cells for u=[u₁, u₂, . . . , u_(n)] must contain    initial condition values for the variables, which can be constants    of formulas of x.-   3. lbcs: ordered list of (n) cells for the left boundary condition    formulas for each system equation. If no boundary condition is    needed for an equation, the cell should have the value ‘NA’.-   4. rbcs: ordered list of (n) cells for the right boundary condition    formulas for each system equation. If no boundary condition is    needed for an equation, the cell should have the value ‘NA’.-   5. domain: the spatial interval limits, and control of output    spatial points similar to format shown in Table 8.-   6. interval: the time interval limits, and control of output time    points similar to format shown in Table 8.-   7. [options]: a list of optional parameters for custom control. They    include supplying an analytic system and boundary conditions    Jacobian matrices, algorithm selection, tolerances values, and    custom algorithm settings.

Based on the flowchart of FIG. 7, PDSOLVE( ) uses Modules-D & E toevaluate the system formulas for the underlining solver algorithm, whichis based on the numerical method of lines described by Schiesser in thebook: “The Numerical Method of Lines”, San Diego, Calif.: AcademicPress, 1991. PDSOLVE( ) generates tabular results in one of two layoutformats for the computed response, which contain two independentvariables for time and space: a transient format which provides a quickview on the transient behavior of system variables at fixed spatialpoints; or a snapshot format which provides a snapshot view of systemvariables distribution at fixed time values. The transient format isdemonstrated in FIG. 8B, where the system variables are reported inrepeated column blocks for each pair of the output time and spacevalues. The snapshot view layout is identical except that the roles oftime and space are interchanged.

The use of PDSOLVE( ) is illustrated in Excel by computing the responseof the following partial differential system:

$\frac{\partial u}{\partial t} = {k\frac{\partial^{2}u}{\partial x^{2}}}$with initial condition u(x, t)=0 at t=0, left boundary condition u(x,t)=100 at x=0, and right boundary condition ∂u(x, t)/∂x=0.25(t−20) atx=1. The problem's spatial domain is x ∈ [0 1] and the time interval ist ∈ [0 1].

Using cells T1, X1, U1, U2, and U3 for system variables t, x, u, u_(x),u_(xx) respectively, the system formula is defined in A1, the leftboundary condition in B1, and the right boundary condition in C1. Avalue of zero is specified for the initial condition in U1, as shown inTable 13.

TABLE 13 A B C K U 1 =K1 * U3 =U1 − 100 =U2 * 0.25 * (T1 − 20) 1 0

Next, the array formula ‘=PDSOLVE (A1, (T1,X1,U1:U3), B1, C1,{0,0.25,0.75,1}, {0,1})’ is evaluated in the allocated Excel rangeG1:K23. It populates the range shown in Table 14 based on the defaulttransient format. FIG. 9C plots the system transient response at thereported spatial points using Excel charts.

TABLE 14 G H I J K 1 X1 0 0.25 0.75 1 2 T1 U1 U1 U1 U1 3 0 0 0 0 0 40.05 0 1.0476E−05 0.0004256 0.002116 5 0.1 0 0.00012165 0.00191450.005946 6 0.15 0 0.00046505 0.0043541 0.010893 7 0.2 0 0.001080830.0076181 0.016767 8 0.25 0 0.00197256 0.0116005 0.02343 9 0.3 00.00312257 0.0162240 0.030791 10 0.35 0 0.00450849 0.0214111 0.03876 110.4 0 0.00610069 0.0270906 0.047261 12 0.45 0 0.00787356 0.03319670.056222 13 0.5 0 0.00980256 0.03967770 0.065588 14 0.55 0 0.011868970.04649278 0.075317 15 0.6 0 0.01405750 0.05360453 0.085366 16 0.65 00.01635493 0.06098080 0.095703 17 0.7 0 0.01875013 0.06859335 0.10629518 0.75 0 0.02122756 0.07640578 0.117104 19 0.8 0 0.02378355 0.084407600.128118 20 0.85 0 0.02640404 0.09256512 0.1393 21 0.9 0 0.029082660.10086286 0.150634 22 0.95 0 0.03181410 0.10928799 0.162106 23 1 00.03458887 0.11781739 0.173691III. Algebraic Equations and Inequalities Solver

The provided Excel spreadsheet function

-   -   NLSOLVE (constraints, variables, nge, [options])        computes an optimal solution to a system of k equations and m−k        inequalities, with variables x=[x₁, x₂, . . . , x_(n)], ordered        in the following form:        ƒ_(i)(x)=0,i=1,k        ƒ_(i)(x)≥0,i=k+1,m

The parameters for NLSOLVE( ) are described below:

-   1. constraints: (m) references to the formulas for the LHS formulas    ƒ_(i).-   2. variables: (n) references to the variable cells. A guess value    must be defined for each variable.-   3. nge: the number of inequalities in the system (m−k). Default is    zero.-   4. [options]: a list of optional parameters for custom control. They    include supplying an analytic constraints Jacobian matrix, algorithm    selection, tolerances values, and custom algorithm settings.

Based on the flowchart of FIG. 7, NLSOLVE( ) uses Modules-D & E, and theLevenberg-Marquardt minimization procedure described in the book“Numerical Recipes in C: The Art of Scientific Computing”, CambridgeUniversity Press, 1992. The Levenberg-Marquardt algorithm finds theoptimal solution to a system of equations and inequalities by minimizingan implicit objective function, which is defined as the sum of squaresof the equations and active inequalities.

NLSOLVE( ) usage in Excel is illustrated by solving the following systemof algebraic equations, which has one solution at (1, 10, 1, 5, 4, 3):x ₃ e ^(−0.1x) ¹ −x ₄ e ^(−0.1x) ² +x ₆ e ^(−0.1x) ⁵ −e ^(−0.1)+5e ⁻¹−3e^(−0.4)=0x ₃ e ^(−0.2x) ¹ −x ₄ e ^(−0.2x) ² +x ₆ e ^(−0.2x) ⁵ −e ^(−0.2)+5e ⁻²−3e^(−0.8)=0x ₃ e ^(−0.3x) ¹ −x ₄ e ^(−0.3x) ² +x ₆ e ^(−0.3x) ⁵ −e ^(−0.3)+5e ⁻³−3e^(−1.2)=0x ₃ e ^(−0.4x) ¹ −x ₄ e ^(−0.4x) ² +x ₆ e ^(−0.4x) ⁵ −e ^(−0.4)+5e ⁻⁴−3e^(−1.6)=0x ₃ e ^(−0.5x) ¹ −x ₄ e ^(−0.5x) ² +x ₆ e ^(−0.5x) ⁵ −e ^(−0.5)+5e ⁻⁵−3e⁻²=0x ₃ e ^(−0.6x) ¹ −x ₄ e ^(−0.6x) ² +x ₆ e ^(−0.6x) ⁵ −e ^(−0.6)+5e ⁻⁶−3e^(−2.4)=0

Using cells X1:X6 for the system variables with a guess value of 1, thesystem equations are defined in cells A1:A6 as shown in Table 15.

TABLE 15 A X 1 =X3 * EXP(−0.1 * X1) − X4 * EXP(−0.1 * X2) + X6 * 1EXP(−0.1 * X5) − EXP(−0.1) + 5 * EXP(−1) − 3 * EXP(−0.4) 2 =X3 *EXP(−0.2 * XI) − X4 * EXP(−0.2 * X2) + X6 * 1 EXP(−0.2 * X5) −EXP(−0.2) + 5 * EXP(−2) − 3 * EXP(−0.8) 3 =X3 * EXP(−0.3 * XI) − X4 *EXP(−0.3 * X2) + X6 * 1 EXP(−0.3 * X5) − EXP(−0.3) + 5 * EXP(−3) − 3 *EXP(−1.2) 4 =X3 * EXP(−0.4 * XI) − X4 * EXP(−0.4 * X2) + X6 * 1EXP(−0.4 * X5) − EXP(−0.4) + 5 * EXP (−4) − 3 * EXP(−1.6) 5 =X3 *EXP(−0.5 * XI) − X4 * EXP(−0.5 * X2) + X6 * 1 EXP(−0.5 * X5) −EXP(−0.5) + 5 * EXP (−5) − 3 * EXP(−2)

Evaluating the formula ‘=NLSOLVE (A1:A6, X1:X6)’ in allocated rangeB1:C6 populates the range with the results shown in Table 16.

TABLE 16 B C 1 X1 =  1 2 X2 = 10 3 X3 =  1 4 X4 =  5 5 X5 =  4 6 X6 =  3Functional Formulation for Optimization Problems

The unique spreadsheet functions introduced thus far form essentialbuilding blocks which enable the construction of a functionalformulation for carrying out a constrained optimization of aparameterized system of differential equations in a spreadsheetapplication. In the following, I present a functional formulation thatprovides the framework for carrying out the optimization process in thespreadsheet application on a computer. In what follows, a bold symbolindicates a vector value. Let u(x, p_(d)) be the response function to asystem of differential equations, where x is an independent variable andp_(d) is a set of design parameters (here x may represent a temporal orspatial variable). u(x, p_(d)) may represent the response of a system ofordinary differential equations with one independent variable x, or to asystem of partial differential equations with two independent variablest and x, by defining u(x, p_(d))≡u(t, x, p_(d)), or alternatively u(t,p_(d))≡u(t, x, p_(d)), where t or x is a specified fixed value for thecorresponding independent variable.

The reader can think of u(x, p_(d)) as an abstract to the tabularresults obtained by any of the spreadsheet functions IVSOLVE( ),BVSOLVE( ), or PDSOLVE( ). In a functional formulation, a pure functionis equivalent to and exchangeable by its value. Accordingly, u(x, p_(d))can be interpreted as providing values for the differential systemvariables u=[u₁, u₂, . . . , u_(n)] at a specified value for x, and fora given configuration of the system design parameters p_(d).

Let ƒ_(i)(u(x, p_(d)), [x], p_(d)) be a criterion function that computesa scalar property from the differential system response u(x, p_(d)), fora specified range of the independent variable [x], and values for thedesign parameters p_(d). For example, ƒ_(i) may simply extract a singlevalue from u(x, p_(d)), or may compute a complex value by applying aprescribed operation, such as integrating a component of u(x, p_(d))over a specified range [x]. Given a target design value τ_(i) for eachcriterion function ƒ_(i), we construct the following system of mconstraints:g _(i)(p _(d))=0i=1,kg _(j)(p _(d))≥0j=k+1,mwhere g_(i) is defined as follows:g _(i)(p _(d))≡ƒ_(i)(u(x,p _(d)),[x],p _(d))−τ_(i) i=1,mLet

${\delta_{j}\left( p_{d} \right)} = \left\{ {{{\begin{matrix}{{0,{{g_{j}\left( p_{d} \right)} \geq 0}}\mspace{14mu}} & {{is}\mspace{14mu}{true}} \\{1,} & {else}\end{matrix}j} = {k + 1}},m} \right.$

The constrained optimization problem can be stated as follows:

${{Minimize}\mspace{14mu}{C\left( p_{d} \right)}} = {{\sum\limits_{i = 1}^{k}{g_{i}\left( p_{d} \right)}^{2}} + {\sum\limits_{j = {k + 1}}^{m}\left\lbrack {{g_{j}\left( p_{d} \right)}{\delta_{j}\left( p_{d} \right)}} \right\rbrack^{2}}}$

In a functional form the minimization of the objective C(p_(d)) isexpressed as the evaluation of a functional of the form:solve(g(p _(d)),p _(d) ,m−k)which maps directly to the spreadsheet function NLSOLVE( ) introducedearlier. NLSOLVE( ) uses Levenberg-Marquardt to find the solution to asystem of equations and inequalities by a minimizing an implicitobjective function equivalent to C(p_(d)) which represents the sum ofthe squares of the constraints values.

The presented functional formulation lays the foundation for athree-step optimization process enabled by the unique spreadsheetfunctions that form the pillars of the functional formulation. The Excelspreadsheet application provides the platform to execute the process ona computer, and the necessary mechanisms to embody the functions. Thethree steps of the process are:

-   -   I. Obtain the response to the parameterized differential system        by a suitable spreadsheet function: IVSOLVE( ), BVSOLVE( ), or        PDSOLVE( ).    -   II. Define constraints on the system response by the aid of        criterion functions.    -   III. Solve the system of constraints for the design parameters        using the spreadsheet function NLSOLVE( ).

Clearly, enabling the three step optimization process is contingent onthe development of criterion spreadsheet functions which enable theconstruction of constraints and provide the link between thedifferential system and the optimizer. Accordingly, a set of criterionspreadsheet functions are developed and described next.

Criterion Worksheet Functions

A criterion function corresponds to ƒ_(i)(u(x, p_(d)), [x], p_(d)) andenables the definition of constraints for the optimization process. Itcomputes a scalar property from the tabular results array of adifferential systems solver, for the purpose of providing a handle onthe system response. A constraint formula then calculates the differencebetween the scalar property and a target design value. One or multipleconstraint formulas are defined, which are then solved for the optimalparameter values using NLSOLVE( ). Three spreadsheet criterion functionsARRAYVAL( ), ODEVAL( ), and PDEVAL( ) are developed for this purpose anddescribed below.

A) Criterion Function: ARRAYVAL( )

The provided Excel spreadsheet function

-   -   ARRAYVAL (data, global_oper, local_oper)        enables the computation of an arbitrary value from a results        array by applying global and local operations to selected data        within the array. The parameters for ARRAYVAL( ) are described        below:

-   1. data: a cell or a range of cells selected form the results array    of a differential system solver.

-   2. global_oper: a reference to a formula to apply to the selected    data. The formula must reference the selected data as VAR1 and    evaluate to a scalar value. For example, to compute the maximum of    data, the formula is defined as ‘=MAX(VAR1)’. As a convenience,    typical global operations such as computing maximum, minimum or mean    can be specified by the string “MAX”, “MIN”, or “MEAN” without the    need to define a formula.

-   3. local_oper: an optional reference to a formula that is applied to    each element of data prior to applying the global operation formula.    The formula must reference elements in data as VAR1. For example, to    apply absolute operator for each element of data, the formula is    defined as ‘=ABS(VAR1)’.

ARRAYVAL( ) uses Modules D & E to evaluate the supplied local and globalformulas for the selected data within the results array.

B) Criterion Function: ODEVAL( )

The provided Excel spreadsheet function

-   -   ODEVAL (range, operand, operation, opr_param)        applies a calculus integration, differentiation, or        interpolation operation to compute a value from the results        array of IVSOLVE( ) or BVSOLVE( ). The operand for the calculus        operation is specified as a formula of one or more system        variable. The operation is applied over a selected range for the        system's independent variable by the aid of a cubic spline curve        fit to the data. The parameters for ODEVAL( ) are described        below:

-   1. range: a range of cells selected from the first column of the    tabular results array (see FIG. 8A) which sets the independent    temporal or spatial variable domain for the calculus operation.

-   2. operand: a reference to a formula that defines the operand for    the calculus operation. The formula references one or more system's    variables. As a convenience, if the operand is a single system's    variable, then the reference to that system's variable can be    specified without the need for a formula.

-   3. operation: a string identifier for the calculus operation to be    performed. The string can be selected from the list below:    -   a. “INTEL” to integrate the operand over the selected range.    -   b. “DERIV” to differentiate the operand at the selected value in        opr_param.    -   c. “INTERP” to interpolate the operand at the selected value in        opr_param.

-   4. opr_param: a value that specifies the differentiation or    interpolation point for the calculus operation. Not required for    integration operation.

FIG. 10 shows the flowchart for the ODEVAL( ) procedure, which definestwo pairs of vectors for the range and corresponding operand values. Therange vector is readily obtained from the first parameter 800. Theoperand data vector is obtained using Modules D & E to evaluate thesupplied operand formula for each point in the range vector 801. Thedata points are then fit to a curve using a standard cubic spline fittechnique 802. Alternative curve fitting models can also be used. Theadvantage of the spline model is that it allows for a direct calculationof the curve integral value, derivative, or interpolated value at anypoint along the curve, using standard procedures, as described by Gao,Zhang and Cao in the article: “Differentiation and numerical Integral ofthe Cubic Spline Interpolation”, in the Journal of Computers, Vol. 6, No10, 2011. The requested operation is performed on the cubic spline curvefit 803, and the computed value is returned 804 to the requester.

C) Criterion Function: PDEVAL( )

The provided Excel spreadsheet function

-   -   PDEVAL (range, operand, block, operation, opr_param)        serves an identical role to that of ODEVAL( ) except that it        applies to the results array obtained by PDSOLVE( ) for a        partial differential system. The additional block parameter        selects the system's variables block in the tabular result array        of PDSOLVE( ). The block value is specified by the value of the        second system's independent variable for the block in the first        row of the results table (see FIG. 8B).        Three-Step Optimization Process for Parameterized Differential        Systems

The presented spreadsheet functions calculate static results whenevaluated manually for their input. On the other hand, they areevaluated dynamically when executed within a functional programrepresenting the three-step optimization process. The criterionfunctions provide the links that connect the differential systemssolvers IVSOLVE( ), BSOLVE( ), and PDSOLVE( ) with the constraintssolver NLSOLVE( ), and enable the operation of the functionalformulation of the spreadsheet functions. The links are achievednaturally, thanks to the design of the formula evaluation graph producedby Module-D, via special handling of criterion functions during theconstruction of graph nodes of type F. This special handling works asfollows: a node of type F that contains a criterion function call isfurther expanded by inserting a child node of type D that contains theunderlining differential system solver call. The formula for the latteris obtained from the first parameter of the criterion function via theExcel API. During the evaluation of the constraints graphs by Module-Eas requested by NLSOLVE( ), nodes of type D are evaluated at the currentvalues for the NLSOLVE( ) system variables, which represent values ofthe design parameters for the underlining differential system.Evaluation of type D nodes is accomplished by a direct call to thedifferential system solver function using the current values of theparameters. The obtained array result for node D is then processed bythe criterion function to compute the requested scalar property, whichrepresents the value for the F node. Module-E then proceeds normally toevaluate remaining graph nodes, and returns the current value of theconstraint formula represented on the graph.

In what follows, I present two examples to demonstrate the merits of thethree-step optimization process utilizing the developed spreadsheetfunctions. Please note the simplicity by which optimal results areobtained for what otherwise would require a complex process. Theseexamples are produced using the developed computer program Add-In forExcel.

Example 1: Customizing a 2^(nd) Order Dynamical System Response

Consider the 2^(nd) order differential equation

${\frac{d^{2}x}{{dt}^{2}} + {2\zeta\; w_{n}\frac{dx}{dt}} + {w_{n}^{2}x}} = 0$which describes a dynamical system with inertia, energy storage, anddissipation components. In this formulation, x(t) represents thedisplacement, w_(n) represents the natural frequency, and ζ the dampingratio. The objective is to find optimal values for w_(n) and ζ such thatthe system has an absolute overshoot value of 2.0 at a peak time of 2.0seconds, in response to an initial displacement condition of one. Theovershoot is defined as the maximum absolute peak value of thedisplacement curve x(t), and the peak time is the time at which theovershoot is attained.Step 1: Solve the differential system with initial values for the designparameters.

In order to use IVSOLVE( ), the 2^(nd) order system is represented astwo first order differential equations:

$\begin{matrix}{\frac{dx}{dt} = v} \\{\frac{dv}{dt} = {{{- 2}\zeta\; w_{n}v} - {w_{n}^{2}x}}}\end{matrix}$

The problem input definition for IVSOLVE( ) is shown in Table 17. Thesystem formulas are defined in B2:B3 using named variables cells t, x,and v, with shown initial conditions. The formulas reference the designparameters zeta and wn, with shown initial values. Two additional namedcells, overshoot and peaktime, are used to hold values for the targetdesign values.

TABLE 17 A B C D 1 Differential system Target design values 2 dx/dt =vovershoot −2.0 3 dv/dt =−2 * zeta * wn * v − wn{circumflex over ( )}2 *x peaktime   2.0 4 System variables/initial conditions Design parameterswith initial values 5 t 0 zeta   0.25 6 x 1 wn   1 7 v 0

Table 18 shows the results obtained by evaluating the formula‘=IVSOLVE(B2:B3, (t,x,v), {0, 12})’ in allocated range I1:K41. The datais plotted in FIG. 11A using Excel charts. The plot shows that thedefault system response exhibits an absolute overshoot over 0.4 atapproximately a peak time of 3.3 when using the initial values of 0.25and 1 of the design parameters zeta and wn.

TABLE 18 I J K 1 t x v 2  0   1   0 3  0.307692   0.95535 −0.28071 4 0.615385   0.833891 −0.49696 5  0.923077   0.657155 −0.6391 6  1.230769  0.448401 −0.70534 7  1.538462   0.230376 −0.70073 8  1.846154  0.023398 −0.63575 9  2.153846 −0.15612 −0.5247 10  2.461538 −0.29644−0.3838 11  2.769231 −0.39094 −0.22958 12  3.076923 −0.43793 −0.07737 13 3.384615 −0.4401   0.05988 14  3.692308 −0.40364   0.172347 15  4−0.33723   0.253769 16  4.307692 −0.25094   0.301494 17  4.615385−0.15511   0.316149 18  4.923077 −0.05944   0.301196 19  5.230769  0.027772   0.26216 20  5.538462   0.100128   0.205861 21  5.846154  0.153446   0.139668 22  6.153846   0.185801   0.070754 23  6.461538  0.197366   0.005509 24  6.769231   0.190099 −0.05091 25  7.076923  0.167319 −0.09486 26  7.384615   0.13322 −0.12428 27  7.692308  0.092385 −0.13868 28  8   0.049327 −0.13896 29  8.307692   0.008121−0.1271 30  8.615385 −0.02792 −0.10586 31  8.923077 −0.05639 −0.07844 32 9.230769 −0.07589 −0.0481 33  9.538462 −0.08601 −0.01789 34  9.846154−0.08719   0.009559 35 10.15385 −0.08061   0.032266 36 10.46154 −0.06796  0.048927 37 10.76923 −0.05119   0.05895 38 11.07692 −0.03235  0.062412 39 11.38462 −0.01339   0.059949 40 11.69231   0.004037  0.052616 41 12   0.018627   0.041749Step 2: Define constraints.

To alter the system response so it attains the target overshoot and peaktime, two equality constraints are defined using the criterion functionsARRAYVAL( ) and ODEVAL( ) as shown in Table 19. Noting (see FIG. 11A)that the displacement curve is negative when the peak is attained,constraint C1 requires that the minimum value of the displacement,(column J2:J41 of Table 18), equals the target overshoot, whileconstraint C2 requires that the interpolated value for the displacementat the target peak time equals the target overshoot. There are otherequally valid constraints that can be used in place of C2 above. Forinstance, we can require that the derivative of the displacement at thetarget peak time vanishes by specifying the constraint ‘=ODEVAL(I2:I41,x, “DERIV”, peaktime)’, or equivalently the interpolated value of thevelocity at the target peak time equals zero by the specifying theconstraint ‘=ODEVAL(I2:I41, v, “INTERP”, peaktime)’. Using either ofthese constraints to replace C2 produces same results.

TABLE 19 C 1 =ARRAYVAL(J2:J41, ″MIN″) − overshoot 2 =ODEVAL(I2:I41, x,″INTERP″, peaktime) − overshootStep 3: Solve the system of constraints for the design parameters.

The system of constraints C1:C2 is solved using NLSOLVE( ) with zeta andwn as variables by the formula ‘=NLSOLVE((C1,C2),(zeta,wn)’ in arrayD1:E3. The results obtained are shown in Table 20. FIG. 11B shows themodified system response, which satisfies the design objectives at theoptimal values for zeta and wn.

TABLE 20 D E 1 zeta 0.45402108 2 wn 1.699024479 3 SSERROR 7.47409E−25

Example 2: Computing Train Propulsion and Travel Time

A frictionless train uses a constant propulsion force to accelerate, butrelies solely on the gravitational pull of the Earth, as well asaerodynamic drag for deceleration. Based on the assumptions of Table 21,we need to compute the required propulsion force and the trip time forthe train to travel between two cities 1000 km apart through a straighttunnel.

TABLE 21 Train mass m = 100,000 (kg) Distance travelled d = 1000,000 (m)Earth radius Re = 6371,000 (m) Gravitational constant g = 10 m/s²Propulsion force F_(p) = constant (N) Unknown value Drag force F_(d)(v)= 0.5 * v²(N) Gravitational force F_(g) (θ) = m * g * cos(θ) (N)Constrained Optimization Model

The forces acting on the train during its motion between the two citiesare depicted in FIG. 12. Applying Newton's second law, the motion forthe train is governed by the 2^(nd) order differential equation:m*{umlaut over (x)}=F _(p) −F _(d)({dot over (x)})+F _(g)(θ(x))with initial conditions x(0)=0, {dot over (x)}(0)=0 at departure City A.Since d/Re<<1, where d is the distance between the two cities and Re isthe radius of the earth, the formula for the angle, θ, spanning the arctravelled by the train as a function of travelled distance x is givenby:

${\theta(x)} = \left\{ \begin{matrix}{\tan^{- 1}\left( \frac{Re}{{d/2} - x} \right)} & {x < {d/2}} \\{\frac{\pi}{2},} & {x = {d/2}} \\{{{\tan^{- 1}\left( \frac{Re}{{d/2} - x} \right)} + \pi},} & {x > {d/2}}\end{matrix} \right.$

Upon arrival at city B, the train comes to a halt, so the finalconditions can be stated as:

$\begin{matrix}{{x\left( t_{f} \right)} = d} \\{{\overset{.}{x}\left( t_{f} \right)} = 0}\end{matrix}$

The final conditions represent constraints on the differential equationmodel for the train's motion. The problem then becomes finding optimalvalues for the unknown propulsion force F_(p) and the final time t_(f)such that the two constraints are satisfied. The answer is found easilyusing the three-step optimization process in Excel spreadsheet, as shownbelow.

Step 1: Solve the differential system with initial values for the designparameters.

Table 22 shows the problem input definition for IVSOLVE( ). The 2^(nd)order model differential equation is represented as two first orderequations in cells B2:B3 using the named cells t, x, and v for thesystem variables, and the named cells Fp, Fd, Fg, and Th for the forceformulas.

TABLE 22 A B C D  1 Differential system Forces formulas  2 dx/dt =v Fd=0.5 * v{circumflex over ( )}2  3 dv/dt =1/m * (Fp − Fd + Fg) Fg =m *g * COS(Th)  4 System variables with Th =IF(x < d/2,ATAN(Re/(d/2 − x)),initial conditions IF(x > d/2,ATAN(Re/(d/2 − x)) + PI( ),PI( )/2))  5 t  0  6 x   0 Constant parameters  7 v   0 g    10  8 Design parameterswith Re 6371000 initial values  9 Fp 1000 d 1000000 10 Tf 2000 m  100000

The system is solved with IVSOLVE( ) for a guess value of 1000 for F_(p)and for a sufficiently long time of 4500 seconds. Table 23 shows theobtained result by evaluating the formula‘=IVSOLVE(B2:B3,(t,x,v),{0,4500})’ in the allocated range J1:L38.

TABLE 23 J K L 1 t x v 2 0 0 0 3 125 6115.463 96.66716 4 250 23618.42180.3757 5 375 50345.74 243.5815 6 500 83614.12 285.3167 7 625 120912.9308.8178 8 750 160244.8 318.6075 9 875 200165.6 318.7776 10 1000239672.8 312.4788 11 1125 278105.9 301.8894 12 1250 315027.7 288.4672 131375 350145.5 273.1784 14 1500 383269.8 256.6382 15 1625 414271.2239.2515 16 1750 443058.6 221.296 17 1875 469577.2 202.9315 18 2000493781.3 184.2855 19 2125 515639.8 165.4439 20 2250 535135 146.4526 212375 552249.8 127.3546 22 2500 566971.2 108.1826 23 2625 579292.288.95507 24 2750 589207.6 69.68509 25 2875 596712.8 50.38592 26 3000601804.6 31.06775 27 3125 604479.3 11.73845 28 3250 604737.5 −7.59494 293375 602579.9 −26.9254 30 3500 598007 −46.2459 31 3625 591022 −65.545432 3750 581623.1 −84.8236 33 3875 569814.6 −104.068 34 4000 555605.2−123.257 35 4125 539003.6 −142.369 36 4250 520018.9 −161.382 37 4375498664.1 −180.264 38 4500 474960.7 −198.957Step 2: Define constraints.

Using ODEVAL( ), we define in Table 24 two constraints, which correspondto the final conditions on the differential model. Constraint C14requires that the interpolated value for x at the final but unknown timet_(f) equal the distance travelled d, while constraint C15 requires thatthe interpolated value for the velocity v at the final time t_(f)vanishes.

TABLE 24 C 14 =ODEVAL(J4:J40, x, ″INTERP″, Tf) − d 15 =ODEVAL(J4:J40, v,″INTERP″, Tf)Step 3: Solve the system of constraints for the design parameters.

Using NLSOLVE( ), the system of constraints C14:C14 is solved with F_(p)and t_(f) as variables by evaluating formula‘=NLSOLVE((C14,C15),(Fp,Tf))’ in array A16:B18. The computed optimalvalues for F_(p) and t_(f) are shown Table 25.

TABLE 25 A B 16 Fp 62648.94 17 Tf  3863.575 18 SSERROR 1E−16

CONCLUSION, RAMIFICATIONS, AND SCOPE

It is remarkable that the presented spreadsheet functions had noparallels in the history of the spreadsheet application since itsinception in 1978. They represent a breakthrough in the utilization ofthe spreadsheet application for a new class of problems beyond thecurrent scope of prior art. Prior art was confined to the limitations ofthe spreadsheet's inherent design, which restricts functions tooperating on constant inputs only. Prior art utilized the conventionalcommand mechanism, which as pointed out earlier, lacks the propertiesrequired to support a functional formulation, and consequently limitsthe scope of problems it can handle.

The advantages of enabling a spreadsheet function to passively evaluateformulas while retaining its distinguished properties are significant.In particular, they enable computing the response and other propertiesof any system that can be modelled by formulas in a spreadsheetapplication by means of a spreadsheet function utilizing an appropriatealgorithm. I have provided several spreadsheet functions that enablecomputing results for the following problem categories:

-   -   Integrals and derivatives of formulas of any order.    -   Systems of differential-algebraic equations.    -   Systems of algebraic equations and inequalities.

The presented spreadsheet functions offer additional unique meritsbeyond enabling their integrated solvers. Evident in the illustratedembodiments, these merits include:

-   -   Simplified inputs. The input parameters for each spreadsheet        function have been reduced to mirror the written model        description. Only minimal, high-level description of the system        model is required.    -   Comprehensive control. Provided spreadsheet functions are        enabled with optional parameters to provide full control over        the underlining solver algorithm. This makes them appealing to        the novice and expert alike.    -   Presentable results. Provided spreadsheet functions produce        results in tabular, customizable formats that blend naturally        with the spreadsheet, and integrate seamlessly with charting        tools, facilitating further data processing.    -   Transparent. A spreadsheet function encompasses its complete        problem definition in an easy to understand format. There is no        hidden logic or settings, no additional inputs, no dialogues,        and no implicit actions needed to evaluate the function.    -   Self-documenting. Unlike commands, a spreadsheet function does        not mutate or modify its inputs. It generates results in a        separate spreadsheet memory assigned to it, which serves to        document the model and its results. In contrast, a command        overwrites its input memory by the results.

More importantly, the spreadsheet functions support a functionalparadigm to carry out an optimization process. As demonstrated by theembodiments, optimizing the response of parameterized system has beenreduced to a direct exercise of stating the goals in three logicalsteps. The reader has seen the simplicity by which definitive resultswere obtained applying the three-step process to constrainedoptimization problems of differential systems. A central facilitator ofthe three-step optimization process is the criterion functions. Thesefunctions enable the definition of objective constraints via a simpleone-step method that encapsulates complex procedures. In addition toproviding the dynamic link between the optimizer and the differentialsystem, the criterion functions offer important advantages by:

-   -   Providing direct and simple means to measure a complex        characteristic of a system response.    -   Enabling the statement of an objective constraint as a simple        formula that measures the difference between an observed        characteristic value and a desired target value.

The aforementioned merits and advantages of the embodiments present aviable commercial opportunity. To realize this opportunity, the uniquespreadsheet functions have been incorporated in an Add-In computerprogram that integrates seamlessly with Excel spreadsheet application,and executes on any computer hardware that supports running Excelapplication.

It is important to emphasize that the presented spreadsheet functions,and any specificities mentioned in my description are by no means to beconstrued as limitations on the scope of the embodiments, but rather asexemplification of one or more embodiments. Many other variations arepossible including:

-   -   Alternative designs for formula representations and evaluation        data structures.    -   Alternative designs for the spreadsheet functions' input        parameters and output formats.    -   Integration of additional algorithms to support more general or        specific forms of the differential and algebraic systems.    -   The incorporation of additional prescribed operations to extend        the utility of the criterion functions.    -   Developing additional spreadsheet functions to support more        models that are not illustrated by the embodiments.

Moreover, the preceding description has focused on the Excel spreadsheetapplication as the preferred platform for the illustrated embodiments.However, Excel should not be construed as the exclusive spreadsheetsuitable for the embodiments. Alternative spreadsheet applications, suchas Google spreadsheet application, can also be utilized to serve as aplatform for the embodiments. Accordingly, the scope should bedetermined not by the embodiments illustrated, but by the appendedclaims and their legal equivalents.

The invention claimed is:
 1. A method for computing a value for anintegral equation, the method comprising: A) displaying and operating atleast one tabular datasheet by executing a spreadsheet softwareapplication on a computer device, each tabular datasheet having aplurality of cells each being designated with a column and rowidentifier, the cells being configured to receive values or formulaexpressions input, evaluate formula expressions, and display output; andB) providing a programming interface to the spreadsheet softwareapplication, the programming interface being configured at least to: B1)receive the identifier for a cell from the cells and retrieve a value ora formula expression from the cell; B2) receive an evaluable formulaexpression, and evaluate its equivalent value, wherein the evaluableformula expression represents an independent textual expression that canbe evaluated to an equivalent value by said programming interface; andC) selecting a one of the cells, referred to as the variable cell, torepresent an integration variable for the integral equation; and D)expressing an integrand of the integral equation by a formula in asecond of the cells, referred to as the integrand formula, which dependson zero or more nested formulas in an additional group of the cells, atleast one of the formulas depends on the variable cell; and E) adding anintegration function to the spreadsheet application having a plural ofinput arguments, the solver function being configured to receive in theinput arguments the identifier for the integrand formula's cell, theidentifier for the variable cell, and values for the integral equation'slimits; and F) evaluating a formula of the integration function in athird of the cells to compute the value of the integral equation anddisplay the value in the third of the cells, wherein computing the valueof the integral equation comprises: F1) utilizing a graph constructoralgorithm communicable with said programming interface for retrievingand representing the integrand formula expression and itsinterdependence on the nested formulas and the variable cell on atree-structured evaluation graph of relational nodes containing anequivalent sequence of formula expressions; F2) utilizing a graphevaluator algorithm communicable with said programming interface forevaluating said evaluation graph, the graph evaluator algorithmcomprising: traversing the relational nodes of said evaluation graph inan order of their interdependence and transforming the formulaexpression in each relational node into an evaluable formula expressionby substituting references to the variable cell by a supplied numericalvalue, and any reference to a traversed child relational node; obtainingthe value of a relational node by evaluating the value of the evaluableformula expression in the relational node via said programminginterface; and aggregating the obtained values of the relational nodesin an order of their interdependence to obtain the value of theintegrand formula expression; F3) utilizing an integration algorithmwhich computes the value of the integral equation based on multipleevaluations of the integrand formula at determined values for theintegration variable by the integration algorithm, wherein theintegration algorithm utilizes said graph evaluator for the evaluationsof the integrand formula, thereby computing the value of the integralequation and displaying the value in the third of the cells not storingor modifying any stored values in rest of the cells of the spreadsheetapplication.
 2. The method of claim 1, wherein the spreadsheet softwareapplication comprises a native spreadsheet software application.
 3. Themethod of claim 1, wherein the spreadsheet software applicationcomprises a web-based spreadsheet software application.
 4. The method ofclaim 1, wherein the integral equation is a proper integral comprisingfinite limits.
 5. The method of claim 1, wherein the integral equationis an improper integral comprising at least one infinite limit.
 6. Themethod of claim 1, wherein the integration function receives inadditional input arguments, the integration algorithm selection and oneor more control parameters.
 7. The method of claim 1, wherein theintegration function receives designated unique names identifiers for atleast the integrand formula's cell and the variable cell in lieu oftheir column and row identifiers.
 8. The method of claim 1, furthercomprising computing a value for a multiple integral equation comprisingtwo or more nested integrals of an integrand, each of the nestedintegrals is with respect to a different integration variable of theintegrand, the method utilizing an equivalent number of nestedintegration function formulas in a group of the cells, wherein the mostinner integration function formula receiving the integrand formula ofthe multiple integral equation and the integration variable and limitsof the most inner nested integral of the multiple integral equation, andwherein each inner integration function formula defining an integrandformula for the next outer integration function formula and receivingthe integration variable and limits of the corresponding next outernested integral of the multiple integral equation, and whereinevaluating the most outer integration function formula computes thevalue of the multiple integral equation.
 9. A computer apparatus forcomputing a value for integral equation, the apparatus comprising: Aprocessor and memory configured to: A) display and operate at least onetabular datasheet by executing a spreadsheet software application on acomputer device, each tabular datasheet having a plurality of cells eachbeing designated with a column and row identifier, the cells beingconfigured to receive values or formula expressions input, evaluateformula expressions, and display output; and B) provide a programminginterface to the spreadsheet software application, the programminginterface being configured at least to: B1) receive the identifier for acell from the cells and retrieve a value or a formula expression fromthe cell; B2) receive an evaluable formula expression, and evaluate itsequivalent value, wherein the evaluable formula expression represents anindependent textual expression that can be evaluated to an equivalentvalue by said programming interface; and C) select a one of the cells,referred to as the variable cell, to represent an integration variablefor the integral equation; and D) express an integrand of the integralequation by a formula in a second of the cells, referred to as theintegrand formula, which depends on zero or more nested formulas in anadditional group of the cells, at least one of the formulas depends onthe variable cell; and E) add an integration function to the spreadsheetapplication having a plural of input arguments, the solver functionbeing configured to receive in the input arguments the identifier forthe integrand formula's cell, the identifier for the variable cell, andvalues for the integral equation's limits; and F) evaluate a formula ofthe integration function in a third of the cells to compute the value ofthe integral equation and display the value in the third of the cells,wherein computing the value of the integral equation comprises: F1)utilizing a graph constructor algorithm communicable with saidprogramming interface for retrieving and representing the integrandformula expression and its interdependence on the nested formulas andthe variable cell on a tree-structured evaluation graph of relationalnodes containing an equivalent sequence of formula expressions; F2)utilizing a graph evaluator algorithm communicable with said programminginterface for evaluating said evaluation graph, the graph evaluatoralgorithm comprising:  traversing the relational nodes of saidevaluation graph in an order of their interdependence and transformingthe formula expression in each relational node into an evaluable formulaexpression by substituting references to the variable cell by a suppliednumerical value, and any reference to a traversed child relational node;obtaining the value of a relational node by evaluating the value of theevaluable formula expression in the relational node via said programminginterface; and aggregating the obtained values of the relational nodesin an order of their interdependence to obtain the value of theintegrand formula expression; F3) utilizing an integration algorithmwhich computes the value of the integral equation based on multipleevaluations of the integrand formula at determined values for theintegration variable by the integration algorithm, wherein theintegration algorithm utilizes said graph evaluator for the evaluationsof the integrand formula, thereby computing the value of the integralequation and displaying the value in the third of the cells not storing2 or modifying any stored values in rest of the cells of the spreadsheetapplication.